I've done this route and it is a bit time consuming but it's very cheap. It can be done in about 1.5-2 hrs depending on traffic.

1)At LGA exit terminal and buy metrocard from vending machine in central terminal. follow signs to the public city buses. Take the "Q33" to the last stop. (cost: $2.25)

2)Follow crowd downstairs to the subway (E,F,M,R trains). Take the "E" train to the last stop (World Trade Center).
(Make sure you are on the platform with trains heading towards "manhattan and brooklyn" and not "queens and jamaica")
(cost: free transfer only if using metrocard)

3)Exit station and walk one block to the "PATH" station. (These are trains going to New Jersey and a different system from NYC subways so you must buy a different ticket from vending machine) Take train going towards newark penn station.
(cost is 1.75)

4)Get off at Newark Penn station. Exit train station and go through exit that says "raymond street" to the bus lanes. Get on the #62 bus to Newark airport which stops at all terminals.
(cost 1.50)

total one way cost is 5.50.

A slightly more expensive alternative method would be to:
-take the Q33 to the last stop like above
-transfer to E train and get off at 34th street penn station.
-exit subway station and walk around the corner to NY's Penn Station (not to be confused with Newark Penn station)
-Buy a combo NJ transit railroad/airtrain ticket to EWR.

Hi - You can take a NY Airport Service bus from LGA to Penn Station NYC for $12. Then catch the NJ Transit train from Penn Station NYC to EWR for $12.50. So that's $24.50 for the two combined. I am sure somebody can figure out how to get from LGA to EWR via subway for a bit cheaper but this is the cheapest way in my opinion.

I've done this route and it is a bit time consuming but it's very cheap. It can be done in about 1.5-2 hrs depending on traffic.

1)At LGA exit terminal and buy metrocard from vending machine in central terminal. follow signs to the public city buses. Take the "Q33" to the last stop. (cost: $2.25)

2)Follow crowd downstairs to the subway (E,F,M,R trains). Take the "E" train to the last stop (World Trade Center).
(cost: free transfer only if using metrocard)

3)Exit station and walk one block to the "PATH" station. (These are trains going to New Jersey and a different system from NYC subways so you must buy a different ticket from vending machine) Take train going towards newark penn station.
(cost is 1.75)

4)Get off at Newark Penn station. Exit train station and go through exit that says "raymond street" to the bus lanes. Get on the #62 bus to Newark airport which stops at all terminals.
(cost 1.50)

total one way cost is 5.50.

A slightly more expensive alternative method would be to:
-take the Q33 to the last stop like above
-transfer to E train and get off at 34th street penn station.
-exit subway station and walk around the corner to NY's Penn Station (not to be confused with Newark Penn station)
-Buy a combo NJ transit railroad/airtrain ticket to EWR.

I've done this route and it is a bit time consuming but it's very cheap. It can be done in about 1.5-2 hrs depending on traffic.

1)At LGA exit terminal and buy metrocard from vending machine in central terminal. follow signs to the public city buses. Take the "Q33" to the last stop. (cost: $2.25)

2)Follow crowd downstairs to the subway (E,F,M,R trains). Take the "E" train to the last stop (World Trade Center).
(Make sure you are on the platform with trains heading towards "manhattan and brooklyn" and not "queens and jamaica")
(cost: free transfer only if using metrocard)

3)Exit station and walk one block to the "PATH" station. (These are trains going to New Jersey and a different system from NYC subways so you must buy a different ticket from vending machine) Take train going towards newark penn station.
(cost is 1.75)

4)Get off at Newark Penn station. Exit train station and go through exit that says "raymond street" to the bus lanes. Get on the #62 bus to Newark airport which stops at all terminals.
(cost 1.50)

total one way cost is 5.50.

A slightly more expensive alternative method would be to:
-take the Q33 to the last stop like above
-transfer to E train and get off at 34th street penn station.
-exit subway station and walk around the corner to NY's Penn Station (not to be confused with Newark Penn station)
-Buy a combo NJ transit railroad/airtrain ticket to EWR.

The minimum cost of a Metrocard purchased at the Metrocard vending machines available at LGA is 4.50. Fortunately, since the PATH turnstiles accept Metrocard as a form of payment, your $4.50 Metrocard will cover the Q33 bus, the transfer to the E train, and the PATH fare from WTC, with 50 cents left over on the card.

If going via NJ Transit from NY Penn Station, it is not necessary to go to street level when connecting from the E train to NJ Transit. There is an exit near the south end of the E train platform at 34th Street station that offers direct access to Penn Station.

4)Get off at Newark Penn station. Exit train station and go through exit that says "raymond street" to the bus lanes. Get on the #62 bus to Newark airport which stops at all terminals.
(cost 1.50)

An alternative here is to walk west on Market Street to Broad St and take go 28. Works well if you have a small carry-on (just like the rest of the trip). The 62 also stops there, but it is more rare, only half of the trips visit the airport and and the 62 makes more stops than go 28.

I am looking for a slightly different route JFK to EWR. Could it be done similarly?

The easiest way to do this is to take the SkyTrain to the subway & take the E train to Manhattan, get off at 34th St Penn Station. ($7.25, about an hour) and then from there, take a NJ Transit train to EWR (I think it's about $12, about 30 minutes, with EWR SkyTrain). KEEP YOUR NJ TRANSIT TICKET as you'll need it to get to the EWR SkyTrain, IIRC. NJ Transit trains leave about every 20-30 minutes from Penn Station, so be aware of that - also, get off at EWR Airport, not the Newark Penn Station stop.

Probably not the cheapest - but one of the fastest...

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I am looking for a slightly different route JFK to EWR. Could it be done similarly?

Guv1976 is absolutely correct with his directions. I would allocate more time for this route just to be safe because it's a bit farther out.

Another alternative in case trains are re-routed due to construction would be to take the Q10 bus to the last stop which is 80th road in Kew Gardens. Walk around the corner to the subway. (again the E train) and get off at WTC for the PATH.

The minimum cost of a Metrocard purchased at the Metrocard vending machines available at LGA is 4.50. Fortunately, since the PATH turnstiles accept Metrocard as a form of payment, your $4.50 Metrocard will cover the Q33 bus, the transfer to the E train, and the PATH fare from WTC, with 50 cents left over on the card.

If going via NJ Transit from NY Penn Station, it is not necessary to go to street level when connecting from the E train to NJ Transit. There is an exit near the south end of the E train platform at 34th Street station that offers direct access to Penn Station.

An alternative here is to walk west on Market Street to Broad St and take go 28. Works well if you have a small carry-on (just like the rest of the trip). The 62 also stops there, but it is more rare, only half of the trips visit the airport and and the 62 makes more stops than go 28.